Problem 2 The number of typhoons in a country follows a Poisson distribution with a mean of 3 typhoons per year. a} 1lintat is the probability that there will be at least two typhoons in this country in two years? {'3 monks) b} After experiencing a typhoon, the country waits for less than [l8 years for the next typhoon. Conditioned on this, what is the probability that this country waits for more than [15 years for the next typhoon? (31' MT'JICS ,1 c) The number of typhoons in another country follows a Poisson distribution with a mean of 4 typhoons per year. Occurrences of typhoons in the two countries are independent. What is tho distribution of tho total number of typhoons in these two countries in a year? Prove your claim. H marks) Problem 3 A group of patients are waiting in the hospital for COVIDlfl test. The test results of all the patients are independent. The probability that a patient tests positive is I? = CLUE). a} 'What is tho probability that at least 2f} patients are tested before a patient tests positive? {2 merits) b] If lll patients are tested, what is the probability that 2 or 3 patients test positive? {2 marks) c] At least how many patients are needed such that the probability that at least 1 patient among these patients tests positive is greater than 0.3? {2 merits) d] If 211'! patients are tested, what is the approximate probability that more than 15 of them test positive? {4' marks)